The statistics of mesoscopic systems and the physical interpretation of extensive and non-extensive entropies

The postulates of thermodynamics lead to the definition of the entropy, which, for a macrosocpic system, is a homogeneous function of order one in the extensive variables and is maximized at equilibrium. For a mesoscopic system, by definition, the size and the contacts with other systems influence its thermodynamic properties and therefore the entropy is not a homogeneous function of order one in the extensive variables. While for macroscopic systems and homogeneous entropies the equilibrium conditions are clearly defined, it is not so clear how the non-extensive entropies should be applied for the calculation of equilibrium properties of mesoscopic systems-for example it is not clear what is the role played by the boundaries and the contacts between the subsystems. We propose here a general definition of the entropy in the equilibrium state, which is applicable to both, macroscopic and mesoscopic systems. This still leaves an apparent ambiguity in the definition of the entropy of a mesoscopic system, but this we recognize as the signature of the anthropomorphic character of the entropy. To exemplify our approach, we analyze four formulas for the entropy and calculate the equilibrium distributions of probabilities by two methods for each.